The Petitioner refers to the person filing the petition with the court. The petition documents typically ask the court to appoint a personal representative (Administrator or Executor) of the Estate. Usually the petitioner is also the individual seeking appointment.
If the decedent passed with a Will (also known as Testate), it's very likely that the Will will name a proposed Executor. If there is a named Executor, this individual has priority and would need to sign a renunciation if anyone else were to be appointed.
If the decedent passed without a Will (Intestate), the next of kin have priority to be appointed. Usually, the consent of all individuals of equal or higher priority is required.